The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation

The response of single degree of freedom systems with quadratic and cubic nonlinearities to a subharmonic excitation is investigated. The method of multiple scales is used to derive two first order ordinary differential equations that govern the evolution of the amplitude and phase of the subharmonic. These equations are used to obtain the steady state solutions and their stability. The results identify two critical values ζ₁ and ζ₂, where ζ₂>ζ₁, for the excitation amplitude f. The value ζ₂ is the threshold for the stability of the trivial solution. When f>ζ₂, subharmonic oscillations of finite amplitude are always excited. When f<ζ₁, subharmonic oscillations cannot be excited. But when ζ₁<f<ζ₂, subharmonic oscillations may or may not be excited, depending on the initial conditions. Also, the method of harmonic balance is applied to a special case of the problem considered. It is shown that, although the method seems straightforward, it can lead to erroneous results if extreme care is not taken in the ordering of the different terms.